Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{-7r^2 - 42r + 189}{-6r^3 + 48r^2 - 90r}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {-7(r^2 + 6r - 27)} {-6r(r^2 - 8r + 15)} $ $ y = \dfrac{7}{6r} \cdot \dfrac{r^2 + 6r - 27}{r^2 - 8r + 15} $ Next factor the numerator and denominator. $ y = \dfrac{7}{6r} \cdot \dfrac{(r - 3)(r + 9)}{(r - 3)(r - 5)}$ Assuming $r \neq 3$ , we can cancel the $r - 3$ $ y = \dfrac{7}{6r} \cdot \dfrac{r + 9}{r - 5}$ Therefore: $ y = \dfrac{ 7(r + 9)}{ 6r(r - 5)}$, $r \neq 3$